Animated Solution for Mathematics - Vector Algebra: Let $\vec{a},\vec{b},\vec{c}$ be three non-coplanar vectors and $\vec{p},\vec{q},\vec{r}$ are vectors defined by the relations $\vec{p}=\frac{\vec{b}\times \vec{c}}{[\vec{a}\vec{b}\vec{c}]},\vec{q}=\frac{\vec{c}\times \vec{a}}{[\vec{a}\vec{b}\vec{c}]},\vec{r}=\frac{\vec{a}\times \vec{b}}{[\vec{a}\vec{b}\vec{c}]}$ then the value of the expression $(\vec{a}+\vec{b})\cdot \vec{p}+(\vec{b}+\vec{c})\cdot \vec{q}+(\vec{c}+\vec{a})\cdot \vec{r}$ is equal to

Welcome, future engineer. Today, we are embarking on a journey into the heart of 3D vector geometry. Often, when we look at vectors, we see arrows in space, but in the JEE Advanced curriculum, vectors are the building blocks of coordinate systems.

Today, we explore the concept of 'reciprocal vectors'—a powerful tool that acts as a mathematical mirror to our standard basis. Imagine you have three non-coplanar vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ starting from a common origin. They define a volume captured by the scalar triple product, denoted as $[\vec{a}\vec{b}\vec{c}]$.

We define a new set of vectors, $\vec{p}$, $\vec{q}$, and $\vec{r}$, which we call the reciprocal system. The definition of $\vec{p}$ is given by:

Notice the numerator $\vec{b}\times \vec{c}$. By the right-hand rule, this cross product is a vector perpendicular to the plane containing $\vec{b}$ and $\vec{c}$. By dividing by the scalar triple product, we normalize this vector to create a 'dual' relationship with the original basis.

Now, let us witness the beauty of this system. When we take the dot product of a base vector with its corresponding reciprocal vector, we find:

This establishes the fundamental identity for the system:

What about the mismatched pairs? Consider $\vec{b}\cdot \vec{p}$. Since $\vec{p}$ is defined as a vector perpendicular to the plane of $\vec{b}$ and $\vec{c}$, it must be perpendicular to $\vec{b}$ itself.

Therefore, $\vec{b}\cdot \vec{p}=0$. This orthogonality is the secret weapon that simplifies complex expressions, as any base vector dotted with a non-corresponding reciprocal vector will always yield zero.

Now, let us tackle the expression:

Using the distributive property of the dot product, we expand the expression into six distinct terms:

Substituting the known values where matching pairs equal $1$ and mismatched pairs equal $0$:

The expression collapses into the final result:

It is a moment of pure mathematical clarity. You didn't need to perform heavy calculations; you only needed to understand the geometric soul of the reciprocal system. Keep this elegance in mind as you continue your preparation.